34 An introduction to the physics of cosmology
so exponentially fast as the hyperbolic trigonometric functions tend to the
exponential.
Because de Sitter space clearly hasH^2 andρin the right ratio for= 1
(obvious, sincek=0), the density parameter in all models tends to unity as the
Hubble parameter tends toH. If we assume that the initial conditions are not fine
tuned (i.e.=O( 1 )initially), then maintaining the expansion for a factor f
produces
= 1 +O(f−^2 ).
This can solve the flatness problem, providedfis large enough. To obtainof
order unity today requires|− 1 |. 10 −^52 at the Grand Unified Theory (GUT)
epoch, and so lnf & 60 e-foldings of expansion are needed; it will be proved
later that this is also exactly the number needed to solve the horizon problem. It
then seems almost inevitable that the process should go to completion and yield
=1 to measurable accuracy today.
2.5.1 Inflation field dynamics
The general concept of inflation rests on being able to achieve a negative-pressure
equation of state. This can be realized in a natural way by quantum fields in the
early universe.
The critical fact we shall need from quantum field theory is that quantum
fields can produce an energy density that mimics a cosmological constant. The
discussion will be restricted to the case of a scalar fieldφ(complex in general, but
often illustrated using the case of a single real field). The restriction to scalar fields
is not simply for reasons of simplicity, but because the scalar sector of particle
physics is relatively unexplored. While vector fields such as electromagnetism
are well understood, it is expected in many theories of unification that additional
scalar fields such as the Higgs field will exist. We now need to look at what these
can do for cosmology.
The Lagrangian density for a scalar field is as usual of the form of a kinetic
minus a potential term:
L=^12 ∂μφ∂μφ−V(φ).
In familiar examples of quantum fields, the potential would be
V(φ)=^12 m^2 φ^2 ,wheremis the mass of the field in natural units. However, it will be better to keep
the potential function general at this stage. As usual, Noether’s theorem gives the
energy–momentum tensor for the field as
Tμν=∂μφ∂νφ−gμνL.From this, we can read off the energy density and pressure:
ρ=^12 φ ̇^2 +V(φ)+^12 (∇φ)^2
p=^12 φ ̇^2 −V(φ)−^16 (∇φ)^2.